A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 83
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Solon and A. J.Bard, /. Am. Chem. Soc, 86, 1926 (1964). Copyright1964, American Chemical Society.]Thus the reverse transition time (for stable R) is always 1/3 that of the forward time(Figure 8.4.1) up to and including r b independent of Do, D R , CQ, and the rate of theelectron transfer (assuming it is sufficiently rapid to show a reverse transition, that is, isnot totally irreversible). The factor of 1/3 means that of the total amount of R generatedduring the forward step (equal to it\lnF mol), only one-third returns to the electrode during the backward step up to r2, with the remainder diffusing into the bulk solution.
Atfirst thought, one might wonder at the independence of r2/t\ on D R . If Z)R is very large,then a larger amount might be expected to diffuse away. However, a large D R also implies that a larger amount will diffuse back during the reverse step, and the mathematicsdemonstrates that this exactly compensates for the diffusion away from the electrode.Thus the T2lh ratio of 1/3 in chronopotentiometry is the analog of ir(2r)/if(r) = 0.293 inpotential step reversal (Section 5.7) and zpc//pa = 1.00 in cyclic voltammetry (Section6.5.1). Expressions can also be written for the potential-time behavior by combining theappropriate kinetic relationship with the equations for CQ(0, t) and C R (0, t). For example, for a nernstian wave, E{)2l5r~ Еф- ^ o r a quasireversible system the separation between ET/4 and E02l5can be used to determine k° (20).MULTICOMPONENT SYSTEMS ANDMULTISTEP REACTIONS (19, 21-23)Consider a solution containing two reducible substances, O\ and O 2 , at concentrations C*and C2, respectively, where the reduction Oi + ще -> Ri occurs first, and then, at morenegative potentials, O2 + n2e —> R2.
Again, assuming semi-infinite linear diffusion, thefollowing response-function equations can be written:[c*[ -fC*(8.5.1)cl/2(8.5.2)8.5 Multicomponent Systems and Multistep Reactions319where ix(s) and i2(s) are the transforms of the individual currents [i\(t) and i2(t)] involvedin the reduction of O\ and O2, respectively. Since the total applied current, /(0» is ii(t) +i2it\ then ~i(s) = ~ix(s) + }2(s), and from (8.5.1) and (8.5.2),^ - C2(0, ,)] =- C l ( 0,(8.5.3)This equation is true at all times.For the period whenjhe potential is not sufficiently negative for O 2 reduction tooccur (i.e., when t ^ т\), C 2 (0, s) = C2/s, i2(s) = 0, and (8.5.3) becomes identical to thesimple equation (8.4.3), so that up to T\ the behavior is unaffected by the presence of O 2 .For t > ть Cx(0, s) = 0, so that (8.5.3) becomes[C *—1l/2FAs(8.5.4)2 ((U)The second transition time (t = т\ + -f-Cт?) occurswhen=the concentration of O2 dropsto zero at the electrode surface, that is, C2(0, s) = 0.
Thus, for a constant current,i(s) = i/s, and (8.5.4) for this time becomes,FAs312(8.5.5)Inversion yields(8.5.6)For example, for the special case WJDJ^C* = « 2 D 2 / 2 C*, т 2 = Зт]. Thus, while in controlledpotential voltammetric methods two substances at equal concentration with equal diffusion coefficients show two waves of equal height, in chronopotentiometry unequaltransition times arise.
The long second transition results from the continued diffusion ofOi to the electrode after т 1? so that only a fraction of the applied current is available forreduction of O 2 (Figure 8.5.1).Similar reasoning shows that for a stepwise process:О + пгеRj + n2e •(8.5.7)(8.5.8)1007.7x1(T6MCd2-1.54x1(T5MPb2--0.2-0.3-0.4-0.5E, V vs.
Ag/AgCI-0.6-0.7-0.8Figure 8.5.1 Consecutive reduction ofPb(II) and Cd(II) at a mercury poolelectrode. Note that if E-t curve isplotted in this manner, it resemblesa voltammogram. [Reprinted withpermission from C. N. Reilley, G. W.Everett, and R. H. Johns, Anal. Chem.,27, 483 (1955). Copyright 1955,American Chemical Society.]320 I» Chapter 8.
Controlled-Current Techniques(а),jf-(b)1TimeFigure 8.5.2 E-t curves for stepwise reduction ofoxygen and uranyl ion at mercury electrode, (a) 1 MLiCl solution saturated with oxygen at 25°C. O 2 +2H2O + 2e -> H 2 O 2 + 2OH"; H 2 O 2 + 2e ->2ОЕГ; T2/TI « 3. (b) 10" 3 M uranyl nitrate in0.1 M KC1 + 0.01 M HC1. U(VI) + e -> U(V);U(V) + 2e -> U(III); тг/т! « 8.
[Reprinted withpermission from T. Berzins and P. Delahay, /. Am.Chem. Soc, 75, 4205 (1953). Copyright 1953,American Chemical Society.]the transition time ratio is given by2n 2n2(8.5.9)Thus for n2 = wi, r 2 = Ът\ (Figure 8.5.2). By use of the response-function principle onecan show (Problem 8.4) that if a current of the form i(t) = fit112 is used, then equal transition times result when n2 = n\.8.6 THE GALVANOSTATIC DOUBLE PULSE METHODIn Section 8.3.4, we saw that the single-pulse galvanostatic method cannot be used forfast electron-transfer reactions (i.e., those with large i 0 ), because the current is primarilynonfaradaic during the initial moments following the application of the current step, and itcontributes mostly to charging Q .
Gerischer and Krause (24) developed a galvanostaticdouble pulse method (GDP), in which two constant-current pulses are applied to the electrode (Figure 8.6.1). The first (large) pulse / b applied for 0 < t < th mainly serves tocharge the double layer to a potential that becomes the point of interrogation by a second,smaller pulse of current /2.
The basic idea is to use the short first pulse (typically 0.5 to 1^ts long) to drive the system to an overpotential that exactly supports the second current.Then 7] will not change when i2 is applied, and there is no significant effect from chargingthe double layer during the second phase. A block diagram of apparatus for this techniqueis shown in Figure 8.6.2.Some faradaic current does flow during the first pulse, and its effects must be takeninto account. The theory of the GDP method was extended along that line by Matsuda, etal.
(25). Valuable instrumental advances were later made by Aoyagi and coworkers(26-28).Figure 8.6.1 Excitation waveform for thegalvanostatic double pulse method.8.6 The Galvanostatic Double Pulse Method321Figure 8.6.2 Block diagram for galvanostatic double-pulse method. A two-electrode cell (i.e.,with a common counter/reference electrode) is placed in a bridge circuit for compensation of thecell ohmic resistance, /fo. The bridge is adjusted with RA = RB, Rc = /fo, RA » RQ SO that thepulse generators produce an essentially constant current through the cell. Provision is sometimesmade in double pulse circuits, however, for a three-electrode cell and potentiostatic control of theworking electrode before the application of the galvanostatic pulse.
For details of such apparatus,see references 26-28.When the ratio of pulse heights (/1Л2) is adjusted properly, and this is determined bytrial and error, the E-t curve following the cessation of the first pulse is horizontal (Figure8.6.3). Under these conditions, the overpotential for a quasireversible one-step, one-electron process is given by (25)l4M nт _ RT , , л-71 - -=г— 1 + Зтг,1/2'32V11Дзтг 1 / 2 и +11 1II(8.6.1)11•_23.••——- "—|"—i1•5Q~ У\ \ \It (0.5 (is/div)20 аIt (0.5 ns/div)Figure 8.6.3 (a) Oveфotential-time traces for galvanostatic double-pulse method for reduction of0.25 mM Hg \+ in 1 M НСЮ4 at an HMDE.
Ratio i2li\ was (1) 7.8, (2) 5.3, (3) 3.2; (2) shows thedesired response for utilization of (8.6.2). (b) Voltage-time traces for constant-current doublepulses applied to equivalent circuit shown. /1 was (1) 7.6, (2) 5.5, (3) 3.3 mA; and /2 = 1 mA.[From M. Kogoma, T. Nakayama, and S. Aoyagi, /. Electroanal. Chem., 34, 123 (1972), withpermission.]322Chapter 8. Controlled-Current Techniquesor, at sufficiently small values of t\,(8.6.2)Thus one can carry out a series of experiments with different pulse widths t\, and plotthe value of 77 at the onset of i2 vs. t]12 to obtain the exchange current from the intercept.Note the similarity between (8.3.10) and (8.6.2).
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